Calculus I
Limits
Derivative
Exercices
Applications
Marginal analysis
© The scientific sentence. 2010
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Calculus I:
Theorems of analysis for the continuous functions
1. Extrema of a function
2. Continuity of a function:
A function f(x) is continuous at x = a if
lim f(x) = f(a)
x → a
A function f(x) is continuous on the interval [a, b] if it
is continuous at each point in the interval.
A closed interval includes its endpoints: [a,b]
An open interval does not include its endpoints [a,b[ or [a,b)
Bounded intervals are finite intervals, that is of finite size.
Conversely, a set which is not bounded is called unbounded.
The set R of all reals is the only interval that is unbounded at both ends.
3. Derivative of f(x)
The derivative of f(x) with respect to x is the function f'(x), defined as,
f'(x) = lim (f(x + h) - f(x) )/ h
h → 0
It is also
f'(x) = lim (f(x) - f(a)) / (x - a)
x → a
df(x)/dx = dy/dx is the Leibniz's notation, f'(x) Lagrange notation,
Newton notation, and Dx is
the Euler notation.
4. Function differentiable
A function f(x) is called differentiable at x = a if f'(a) exists, and
f(x)is called differentiable on an interval if the derivative exists for each point in that interval.
Theorem
If f(x) is differentiable at x = a then f(x) is continuous at x = a.
5. Extreme value theorem:
if a fuction f(x) is continuous on a closed interval [a,b], then
f(x) has both absolute maximun and minimum value on [a,b].
f cont [a,b] it exists c, d in [a,b]: f(c) ≤ f(x) ≤ f(d) for all x in [a,b]
In other words,
if a fuction f(x) is continuous on a closed interval [a,b], then
there are two numbers a < c, d < b so that f(c) is an absolute maximum for
the function and f(d) is an absolute minimum for the function.
Definition
f(x) has :
1. an absolute maximum at x = c if f(x) ≤ f(c) for every x in the domain considered.
2. a relative maximum at x = c if f(x) ≤ f(c) for every x in some open interval around x = c .
3. an absolute minimum at x = c if f(x) ≥ f(c) for every x in the domain considered.
4. a relative minimum at x = c if f(x) ≥ f(c)for every x in some open interval around x = c.
We say absolute or global, relative or local.
6. Intermediate Value Theorem
if f(x) is continuous on [a, b] and M is any number between
f(a) and f(b), then there exists a number c such that:
f(c) = M , and a < c < b
This theorem is used to justify that a function has a root in
an inerval.
7. Mean Value Theorem
mvt.png
if a function f(x) is
continuous on the closed interval [a,b],
differentiable on the open interval (a,b),
then there is a number c such that a < c < b and
f'(c) = (f(b) - f(a))/(b - a)
8. Rolle’s Theorem
if a function f(x) is
continuous on the closed interval [a,b,
differentiable on the open interval (a,b),
f(a) = f(b)
then there is a number c such that a < c < b and
f'(c) = 0 (or f(x) has a critical point in (a,b)).
9. Fermat’s Theorem
if f(x) has a relative extrema at x = c and f'(c) exists then
x = c is a critical point of f(x), such that f'(c) = 0.
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